Gallai-Colorings of Triples and 2-Factors of <svg     style="vertical-align:-4.47127pt;width:29.575001px;" id="M1" height="21.0875" version="1.1" viewBox="0 0 29.575001 21.0875" width="29.575001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.022,-0,0,-.022,.062,15.462)"><path id="x212C" d="M895 660l7 -19q-44 -20 -72 -40q72 -59 72 -138q0 -51 -45 -94q-43 -43 -108 -51v-2q43 -18 66.5 -53.5t23.5 -76.5q0 -45 -26 -89.5t-70 -74.5q-49 -33 -112 -33q-42 0 -68 23t-26 55q0 51 36.5 86t101.5 46l4 -23q-43 -5 -74 -33t-31 -61q0 -20 14.5 -34t35.5 -14
q36 0 61 15q36 21 57 63t21 91q0 70 -46 95q-24 -6 -49 -6q-38 0 -47 19q-158 -326 -414 -326q-76 0 -125 35q-48 36 -48 87q0 31 17 51.5t42 20.5q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 161 53
q67 54 151 202q70 124 134 203q64 77 141 127q-82 56 -210 56q-91 0 -178 -32q-85 -32 -139 -92q-55 -61 -55 -137q0 -48 30 -77t81 -29q84 0 144 75q61 76 65 196h20q0 -125 -64 -213t-170 -88q-65 0 -106 38t-41 105q0 64 39 120.5t103 91.5q134 72 269 72q144 0 244 -67
q40 22 87 40zM841 485q0 54 -40 95q-43 -36 -79 -89t-83 -143q7 4 23 4q25 0 55 -14q60 4 92 49q32 44 32 98z" /></g>

			<g transform="matrix(.016,-0,0,-.016,21.363,20.838)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g>

		
	
</svg>

<table>Diagram of the stars <svg height="15.6875" id="M249" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 12.15 15.6875" width="12.15" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68
l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" id="x1D446"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.225,15.475)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
</svg> with base <svg height="16.6625" id="M250" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 20.7125 16.6625" width="20.7125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z" id="x1D453"></path></g><g transform="matrix(.017,-0,0,-.017,10.976,12.162)"><path d="M546 430q-14 -33 -37.5 -140t-38.5 -209q-29 -195 -156 -289q-29 -21 -70 -37t-78 -16q-62 0 -102.5 37t-40.5 69q0 27 26 48q19 15 25 -3q13 -43 46.5 -69t87.5 -26q55 0 83 21q29 22 50 65t43 140q14 64 29 163h-2q-55 -81 -149 -152q-59 -44 -102 -44q-23 0 -43.5 30
t-20.5 85q0 74 32.5 145.5t84.5 117.5q40 35 100 58.5t117 23.5q21 0 65 -8q30 -6 44 -6zM456 386q-40 16 -81 16q-29 0 -64 -13q-42 -17 -85 -97q-19 -36 -31.5 -86t-12.5 -86q0 -64 30 -64q20 0 52.5 21t60.5 52q74 81 95 123q26 62 36 134z" id="x1D454"></path></g>
</svg> (red) and the corresponding hypergraphs <svg height="15.4" id="M251" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 18.9 15.4" width="18.9" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28
h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" id="x1D43B"></path></g>
<g transform="matrix(.012,-0,0,-.012,14.975,15.187)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
</svg> (blue).</table>

International Journal of Combinatorics

fig1

Figure 1

Figure 1: Gallai-Colorings of Triples and 2-Factors of <svg     style="vertical-align:-4.47127pt;width:29.575001px;" id="M1" height="21.0875" version="1.1" viewBox="0 0 29.575001 21.0875" width="29.575001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.022,-0,0,-.022,.062,15.462)"><path id="x212C" d="M895 660l7 -19q-44 -20 -72 -40q72 -59 72 -138q0 -51 -45 -94q-43 -43 -108 -51v-2q43 -18 66.5 -53.5t23.5 -76.5q0 -45 -26 -89.5t-70 -74.5q-49 -33 -112 -33q-42 0 -68 23t-26 55q0 51 36.5 86t101.5 46l4 -23q-43 -5 -74 -33t-31 -61q0 -20 14.5 -34t35.5 -14
q36 0 61 15q36 21 57 63t21 91q0 70 -46 95q-24 -6 -49 -6q-38 0 -47 19q-158 -326 -414 -326q-76 0 -125 35q-48 36 -48 87q0 31 17 51.5t42 20.5q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 161 53
q67 54 151 202q70 124 134 203q64 77 141 127q-82 56 -210 56q-91 0 -178 -32q-85 -32 -139 -92q-55 -61 -55 -137q0 -48 30 -77t81 -29q84 0 144 75q61 76 65 196h20q0 -125 -64 -213t-170 -88q-65 0 -106 38t-41 105q0 64 39 120.5t103 91.5q134 72 269 72q144 0 244 -67
q40 22 87 40zM841 485q0 54 -40 95q-43 -36 -79 -89t-83 -143q7 4 23 4q25 0 55 -14q60 4 92 49q32 44 32 98z" /></g>

			<g transform="matrix(.016,-0,0,-.016,21.363,20.838)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g>

		
	
</svg> 